Title: Quantitative estimates on singularities
of minimal hypersurfaces
Abstract: We will discuss the
occasionally unavoidable presence of singularities on
minimal hypersurfaces in high dimensional ambient spaces and
estimates on its size. This is seemingly an analysis problem
but the variational notion of minimal hypersurfaces plays a
much more important role than its defining PDE. The proof
relies simply on coverings arguments by suitable open sets
and we will go over the main ideas and consequences.
Date: Tuesday, 25th March 2025,
09:00-10:00 TST (GMT+8) , online on Zoom
Title: Minimal hypersurfaces: bubble convergence and index
Abstract:
The regularity theories of Schoen--Simon--Yau and Schoen--Simon for stable minimal hypersurfaces are foundational in geometric analysis. Using this regularity theory, in low dimensions, Chodosh--Ketover--Maximo, and Buzano--Sharp, studied singularity formation along sequences of minimal hypersurfaces through a bubble analysis.
I will review this background, before talking about my recent work in this bubble analysis theory. In particular I will show how to obtain upper semicontinuity of index plus nullity along a bubble converging sequence of minimal hypersurfaces.